Low Reynolds number swimming in complex environments

The study of swimming micro-organisms has been of interest not just to biologists, but also to fluid dynamicists for over a century. As they are rarely in isolation, much interest has been focused on the study of the swimmers’ interaction with their environment. By virtue of the typically small sizes of these organisms and their swimming protocols, the characteristic Reynolds number of the motion of the fluid around them is small. Hence they reside in a Stokes flow regime where viscous forces dominate inertial effects and where far-field interactions (e.g. with nearby walls) can have a significant effect on the swimmer’s dynamical evolution. This thesis provides a detailed investigation of idealised models of low Reynolds number swimmers in a variety of wall-bounded fluid domains. Our approach employs a combination of analytical and numerical techniques. A simple two-dimensional point singularity is used to model a swimmer. We first study its dynamics when placed in the half-plane above an infinite no-slip wall and find it to be in qualitative agreement with numerical and experimental studies. The success of the model in this case encourages its use to study the swimmer’s dynamics in more complicated domains. Specifically, we next explore the dynamics of the same swimmer above an infinite straight wall with a single gap, or orifice. Using techniques of complex analysis and conformal mapping theory, a dynamical system governing the swimmer’s motion is explicitly derived. This analysis is then extended to the case in which the swimmer evolves near an infinite straight wall with two gaps. We are also interested in how the presence of background flows can affect the swimmer’s dynamics in these confined geometries. We therefore employ the same techniques of complex analysis and conformal mappings to find analytical expressions for pressure-driven flows near a wall with either one or two gaps. We then extend this to find new solutions for the shear flows and stagnation point flows in the same geometry. The effect of a background shear flow on the swimmer’s dynamics is then explored. Finally, while there have been a number of studies of Stokes flows within domains which are simply connected, the doubly connected analogues are rather rare. By building upon the analytical techniques presented in this thesis, we present numerical solutions to such problems, including that of theWeis-Fogh mechanism in the low Reynolds number regime

A two-dimensional model of low-Reynolds number swimming beneath a free surface

Biological organisms swimming at low Reynolds number are often influenced by the presence of rigid boundaries and soft interfaces. In this paper we present an analysis of locomotion near a free surface with surface tension. Using a simplified two-dimensional singularity model, and combining a complex variable approach with conformal mapping techniques, we demonstrate that the deformation of a free surface can be harnessed to produce steady locomotion parallel to the interface. The crucial physical ingredient lies in the nonlinear hydrodynamic coupling between the disturbance flow created by the swimmer and the free boundary problem at the fluid surface

Hydrodynamic bound states of a low-Reynolds-number swimmer near a gap in a wall

The motion of an organism swimming at low Reynolds number near an infinite straight wall with a finite-length gap is studied theoretically within the framework of a two-dimensional model. The swimmer is modelled as a point singularity of the Stokes equations dependent on a single real parameter. A dynamical system governing the position and orientation of the model swimmer is derived in analytical form. The dynamical system is studied in detail and a bifurcation analysis performed. The analysis reveals,inter alia, the presence of stable equilibrium points in the gap region as well as Hopf bifurcations to periodic bound states. The reduced-model system also exhibits a global gluing bifurcation in which two symmetric periodic orbits merge at a saddle point into symmetric ‘figure-of-eight’ bound states having more complex spatiotemporal structure. The additional effect of a background shear is also studied and is found to introduce new types of bound state. The analysis allows us to make theoretical predictions as to the possible behaviour of a low-Reynolds-number swimmer near a gap in a wall. It offers insights into the use of gaps or orifices as possible control devices for such swimmers in confined environments.</jats:p

Low Reynolds number swimming in complex environments

The study of swimming micro-organisms has been of interest not just to biologists, but also to fluid dynamicists for over a century. As they are rarely in isolation, much interest has been focused on the study of the swimmers’ interaction with their environment. By virtue of the typically small sizes of these organisms and their swimming protocols, the characteristic Reynolds number of the motion of the fluid around them is small. Hence they reside in a Stokes flow regime where viscous forces dominate inertial effects and where far-field interactions (e.g. with nearby walls) can have a significant effect on the swimmer’s dynamical evolution. This thesis provides a detailed investigation of idealised models of low Reynolds number swimmers in a variety of wall-bounded fluid domains. Our approach employs a combination of analytical and numerical techniques. A simple two-dimensional point singularity is used to model a swimmer. We first study its dynamics when placed in the half-plane above an infinite no-slip wall and find it to be in qualitative agreement with numerical and experimental studies. The success of the model in this case encourages its use to study the swimmer’s dynamics in more complicated domains. Specifically, we next explore the dynamics of the same swimmer above an infinite straight wall with a single gap, or orifice. Using techniques of complex analysis and conformal mapping theory, a dynamical system governing the swimmer’s motion is explicitly derived. This analysis is then extended to the case in which the swimmer evolves near an infinite straight wall with two gaps. We are also interested in how the presence of background flows can affect the swimmer’s dynamics in these confined geometries. We therefore employ the same techniques of complex analysis and conformal mappings to find analytical expressions for pressure-driven flows near a wall with either one or two gaps. We then extend this to find new solutions for the shear flows and stagnation point flows in the same geometry. The effect of a background shear flow on the swimmer’s dynamics is then explored. Finally, while there have been a number of studies of Stokes flows within domains which are simply connected, the doubly connected analogues are rather rare. By building upon the analytical techniques presented in this thesis, we present numerical solutions to such problems, including that of theWeis-Fogh mechanism in the low Reynolds number regime.EThOS - Electronic Theses Online ServiceGBUnited Kingdo